Lecture 27
•Review
• Phasor voltage-current relations for circuit
elements
•Impedance and admittance
•Steady-state sinusoidal analysis
• Examples
•Related educational modules:
–Section 2.7.3, 2.7.4
Phasor voltage-current relations
IC
IL
IR
+
+
+
R
VR
j L
VL
-
-
V R  R IR
Imaginary
V L  jL  I L
1
j C
C
-
VC 
1
IC
j C
Imaginary
Imaginary
V  RI
V
I
V  j L  I
Real
I
I
Real
Real
V 
1
I
j C
Impedance
• Define the impedance, Z , of a circuit as:
V
Z
I
V  IZ
• Notes:
• Impedance defines the relationship between the voltage
and current phasors
• The above equations are identical in form to Ohm’s Law
• Units of impedance are ohms ()
Impedance – continued
• Impedance is a complex number
Z  R  jX
• Where
• R is called the resistance
• X is called the reactance
• Impedance is not a phasor
• There is no sinusoidal waveform it is describing
Circuit element impedances
• Our phasor circuit element voltage-current relations
can all be written in terms of impedances
ZR  R
Z L  jL
1
ZC 
j C
Admittance
• Admittance is the inverse of impedance
1
Y
Z
• Admittance is a complex number
Y  G  jB
• Where
• G is called the conductance
• B is called the susceptance
Why are impedance and admittance useful?
• The analysis techniques we used for time domain
analysis of resistive networks are applicable to phasor
circuits
• E.g. KVL, KCL, circuit reduction, nodal analysis, mesh
analysis, Thevenin’s and Norton’s Theorems…
• To apply these methods:
• Impedances are substituted for resistance
• Phasor voltages, currents are used in place of time
domain voltages and currents
Steady state sinusoidal (AC) analysis
• KVL, KCL apply directly to phasor circuits
• Sum of voltage phasors around closed loop is zero
• Sum of current phasors entering a node is zero
• Circuit reduction methods apply directly to phasor
circuits
• Impedances in series, parallel combine exactly like
resistors in series, parallel
• Voltage, current divider formulas apply to phasor
voltages, currents
AC analysis – continued
• Nodal, mesh analyses apply to phasor circuits
• Node voltages and mesh currents are phasors
• Impedances replace resistances
• Superposition applies in frequency domain
• If multiple signals exist at different frequencies,
superposition is the only valid frequency domain approach
• Summation of individual contributions must be done in
the time domain (unless all contributions have same
frequency)
AC analysis – continued
• Thévenin’s and Norton’s Theorems apply to phasor
circuits
• voc and isc become phasors ( V OC and I SC )
• The Thévenin resistance, RTH, becomes an impedance,
• Maximum power transfer:
• To provide maximum AC power to a load, the load
impedance must be the complex conjugate of the
Thévenin impedance
Z TH
Example 1
• Determine i(t) and v(t), if vs(t) = 100cos(2500t)V
Example 2
• In the circuit below, vs(t) = 5cos(3t). Determine:
(a) The equivalent impedance seen by the source
(b) The current delivered by the source
(c) The current i(t) through the capacitor
Example 2 – part (a)
(a) Determine the impedance seen by the source
Example 2 – part (b)
(b) Determine current delivered by the source
Example 2 – part (c)
(c) Determine current i(t) through the capacitor
Example 3
• Use nodal analysis to determine the current phasors I C and I R
if I S  10 20
;
• On previous slide:
– Set up reference node, independent node
– Write KCL at independent node
– Solve for node voltage
Example 3 – continued
Example 3 – continued again
• What are ic(t) and iR(t)?
• What are ic(t) and iR(t) if
the frequency of the input
current is 5000 rad/sec?
Example 3 – revisited
• Can example 3 be done more easily?
Example 4
• Use mesh analysis to determine V .
Example 4 – continued